0

I'm trying to write a version of the change of variable theorem for the case of locally integrable functions on open subsets of $\mathbb R^n$.

Statement: Let $U, V\subseteq \mathbb R^n$ be open sets and $\phi:U\longrightarrow V$ be a $C^1$ diffeomorphism and let $f:V\longrightarrow \mathbb C$.

(a) If $f:V\longrightarrow \mathbb C$ is measurable then $f\circ \phi:U\longrightarrow \mathbb C$ is measurable;

(b) The map $x\mapsto |J\phi(x)|(f\circ \phi)(x)$ is locally integrable on $U$ if and only if $f$ is locally integrable on $V$. Furthermore, for every $A\subseteq U$ compact $$\int_{\phi(A)}f(y)\ dy=\int_{A} (f\circ \phi)(x)|J\phi(x)|\ dx$$ holds.

Recall $f$ is locally integrable in $U$ if $f$ is integrable in every compact set $K\subset U$.

My questions are:

(i) The above statement is really true?

(ii) If it is true, can I deduce it from the usual change of variables theorem?

(iii) Why is the hypothesis $\phi\in C^1(U, V)$ important?

PtF
  • 9,895
  • I corrected the spelling of "diffeomorphism" in this and another post of yours. Also, please be aware that there is no such thing as "the usual change of variables theorem". What is usual for you may be unusual for someone who studied from a different book. –  Aug 10 '14 at 21:03

1 Answers1

1

To prove (a), we must show that for every open set $W\subset U$ the preimage $(f\circ \phi)^{-1}(W)$ is measurable. This preimage is $\phi^{-1}(f^{-1}(W))$, and here $f^{-1}(W)$ is measurable. Write $f^{-1}(W)$ as the union of a Borel set and a null set. Under $\phi^{-1}$, Borel sets go to Borel sets (proved here) and null sets go to null sets, because $\phi^{-1}$ is locally Lipschitz. Thus, $\phi^{-1}(f^{-1}(W))$ is also the union of a Borel set and a null set, hence measurable.

(b) The set $\phi(A)$ is a compact subset of $V$. Let $g=f\chi_{\phi(A)}$. This is an integrable function. The identity you want is now $$ \int_{V}g(y)\ dy=\int_{U} (g\circ \phi)(x)|J\phi(x)|\ dx $$ which is the usual (?) change of variable formula. On both sides we integrate over compact subsets, so everything is integrable, not just locally.

The assumption $C^1$ can be weakened to "Lipschitz". And if we are willing to count multiplicities, then $\phi$ need not be bijective. The book by Evans and Gariepy has such a form of the change of variable formula.

Community
  • 1